LMFDB - Embedded newform 9025.2.a.bi.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9025,2,Mod(1,9025)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9025, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("9025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9025 = 5^{2} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9025.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$72.0649878242$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{20})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 5 $ x^4 - 5*x^2 + 5
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1805)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.17557$ of defining polynomial
Character	$\chi$	$=$	9025.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.07768 q^{3} -2.00000 q^{4} +0.618034 q^{7} +6.47214 q^{9} +O(q^{10})$	$q+3.07768 q^{3} -2.00000 q^{4} +0.618034 q^{7} +6.47214 q^{9} -5.85410 q^{11} -6.15537 q^{12} -3.07768 q^{13} +4.00000 q^{16} +2.85410 q^{17} +1.90211 q^{21} -5.47214 q^{23} +10.6861 q^{27} -1.23607 q^{28} +3.80423 q^{29} +1.62460 q^{31} -18.0171 q^{33} -12.9443 q^{36} -8.33499 q^{37} -9.47214 q^{39} +11.5842 q^{41} +7.38197 q^{43} +11.7082 q^{44} -4.70820 q^{47} +12.3107 q^{48} -6.61803 q^{49} +8.78402 q^{51} +6.15537 q^{52} -5.15131 q^{53} -4.25325 q^{59} -7.23607 q^{61} +4.00000 q^{63} -8.00000 q^{64} +7.33094 q^{67} -5.70820 q^{68} -16.8415 q^{69} -0.171513 q^{71} +11.0000 q^{73} -3.61803 q^{77} -13.5923 q^{79} +13.4721 q^{81} -10.7082 q^{83} -3.80423 q^{84} +11.7082 q^{87} -8.22899 q^{89} -1.90211 q^{91} +10.9443 q^{92} +5.00000 q^{93} -8.33499 q^{97} -37.8885 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{4} - 2 q^{7} + 8 q^{9}+O(q^{10}) $ 4 * q - 8 * q^4 - 2 * q^7 + 8 * q^9	$ 4 q - 8 q^{4} - 2 q^{7} + 8 q^{9} - 10 q^{11} + 16 q^{16} - 2 q^{17} - 4 q^{23} + 4 q^{28} - 16 q^{36} - 20 q^{39} + 34 q^{43} + 20 q^{44} + 8 q^{47} - 22 q^{49} - 20 q^{61} + 16 q^{63} - 32 q^{64} + 4 q^{68} + 44 q^{73} - 10 q^{77} + 36 q^{81} - 16 q^{83} + 20 q^{87} + 8 q^{92} + 20 q^{93} - 80 q^{99}+O(q^{100}) $ 4 * q - 8 * q^4 - 2 * q^7 + 8 * q^9 - 10 * q^11 + 16 * q^16 - 2 * q^17 - 4 * q^23 + 4 * q^28 - 16 * q^36 - 20 * q^39 + 34 * q^43 + 20 * q^44 + 8 * q^47 - 22 * q^49 - 20 * q^61 + 16 * q^63 - 32 * q^64 + 4 * q^68 + 44 * q^73 - 10 * q^77 + 36 * q^81 - 16 * q^83 + 20 * q^87 + 8 * q^92 + 20 * q^93 - 80 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0		−	1.00000i	$-0.5\pi$
$2$	0	0			1.00000i	$0.5\pi$
$3$	3.07768	1.77690	0.888451	−	0.458972i	$-0.151782\pi$
$3$	3.07768	1.77690	0.888451	+	0.458972i	$0.151782\pi$
$4$	−2.00000	−1.00000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.618034	0.233595	0.116797	−	0.993156i	$-0.462737\pi$
$7$	0.618034	0.233595	0.116797	+	0.993156i	$0.462737\pi$
$8$	0	0
$9$	6.47214	2.15738
$10$	0	0
$11$	−5.85410	−1.76508	−0.882539	−	0.470239i	$-0.844168\pi$
$11$	−5.85410	−1.76508	−0.882539	+	0.470239i	$0.844168\pi$
$12$	−6.15537	−1.77690
$13$	−3.07768	−0.853596	−0.426798	−	0.904347i	$-0.640358\pi$
$13$	−3.07768	−0.853596	−0.426798	+	0.904347i	$0.640358\pi$
$14$	0	0
$15$	0	0
$16$	4.00000	1.00000
$17$	2.85410	0.692221	0.346111	−	0.938194i	$-0.387502\pi$
$17$	2.85410	0.692221	0.346111	+	0.938194i	$0.387502\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	0	0
$21$	1.90211	0.415075
$22$	0	0
$23$	−5.47214	−1.14102	−0.570510	−	0.821291i	$-0.693254\pi$
$23$	−5.47214	−1.14102	−0.570510	+	0.821291i	$0.693254\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	10.6861	2.05655
$28$	−1.23607	−0.233595
$29$	3.80423	0.706427	0.353214	−	0.935543i	$-0.385089\pi$
$29$	3.80423	0.706427	0.353214	+	0.935543i	$0.385089\pi$
$30$	0	0
$31$	1.62460	0.291787	0.145893	−	0.989300i	$-0.453394\pi$
$31$	1.62460	0.291787	0.145893	+	0.989300i	$0.453394\pi$
$32$	0	0
$33$	−18.0171	−3.13637
$34$	0	0
$35$	0	0
$36$	−12.9443	−2.15738
$37$	−8.33499	−1.37026	−0.685132	−	0.728419i	$-0.740256\pi$
$37$	−8.33499	−1.37026	−0.685132	+	0.728419i	$0.740256\pi$
$38$	0	0
$39$	−9.47214	−1.51676
$40$	0	0
$41$	11.5842	1.80915	0.904573	−	0.426318i	$-0.140190\pi$
$41$	11.5842	1.80915	0.904573	+	0.426318i	$0.140190\pi$
$42$	0	0
$43$	7.38197	1.12574	0.562870	−	0.826546i	$-0.309697\pi$
$43$	7.38197	1.12574	0.562870	+	0.826546i	$0.309697\pi$
$44$	11.7082	1.76508
$45$	0	0
$46$	0	0
$47$	−4.70820	−0.686762	−0.343381	−	0.939196i	$-0.611572\pi$
$47$	−4.70820	−0.686762	−0.343381	+	0.939196i	$0.611572\pi$
$48$	12.3107	1.77690
$49$	−6.61803	−0.945433
$50$	0	0
$51$	8.78402	1.23001
$52$	6.15537	0.853596
$53$	−5.15131	−0.707587	−0.353793	−	0.935324i	$-0.615108\pi$
$53$	−5.15131	−0.707587	−0.353793	+	0.935324i	$0.615108\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−4.25325	−0.553727	−0.276863	−	0.960909i	$-0.589295\pi$
$59$	−4.25325	−0.553727	−0.276863	+	0.960909i	$0.589295\pi$
$60$	0	0
$61$	−7.23607	−0.926484	−0.463242	−	0.886232i	$-0.653314\pi$
$61$	−7.23607	−0.926484	−0.463242	+	0.886232i	$0.653314\pi$
$62$	0	0
$63$	4.00000	0.503953
$64$	−8.00000	−1.00000
$65$	0	0
$66$	0	0
$67$	7.33094	0.895617	0.447808	−	0.894130i	$-0.352205\pi$
$67$	7.33094	0.895617	0.447808	+	0.894130i	$0.352205\pi$
$68$	−5.70820	−0.692221
$69$	−16.8415	−2.02748
$70$	0	0
$71$	−0.171513	−0.0203549	−0.0101774	−	0.999948i	$-0.503240\pi$
$71$	−0.171513	−0.0203549	−0.0101774	+	0.999948i	$0.503240\pi$
$72$	0	0
$73$	11.0000	1.28745	0.643726	−	0.765256i	$-0.277388\pi$
$73$	11.0000	1.28745	0.643726	+	0.765256i	$0.277388\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−3.61803	−0.412313
$78$	0	0
$79$	−13.5923	−1.52925	−0.764627	−	0.644473i	$-0.777077\pi$
$79$	−13.5923	−1.52925	−0.764627	+	0.644473i	$0.777077\pi$
$80$	0	0
$81$	13.4721	1.49690
$82$	0	0
$83$	−10.7082	−1.17538	−0.587689	−	0.809087i	$-0.699962\pi$
$83$	−10.7082	−1.17538	−0.587689	+	0.809087i	$0.699962\pi$
$84$	−3.80423	−0.415075
$85$	0	0
$86$	0	0
$87$	11.7082	1.25525
$88$	0	0
$89$	−8.22899	−0.872272	−0.436136	−	0.899881i	$-0.643653\pi$
$89$	−8.22899	−0.872272	−0.436136	+	0.899881i	$0.643653\pi$
$90$	0	0
$91$	−1.90211	−0.199396
$92$	10.9443	1.14102
$93$	5.00000	0.518476
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−8.33499	−0.846290	−0.423145	−	0.906062i	$-0.639074\pi$
$97$	−8.33499	−0.846290	−0.423145	+	0.906062i	$0.639074\pi$
$98$	0	0
$99$	−37.8885	−3.80794
$100$	0	0
$101$	−1.90983	−0.190035	−0.0950176	−	0.995476i	$-0.530291\pi$
$101$	−1.90983	−0.190035	−0.0950176	+	0.995476i	$0.530291\pi$
$102$	0	0
$103$	−9.06154	−0.892860	−0.446430	−	0.894819i	$-0.647305\pi$
$103$	−9.06154	−0.892860	−0.446430	+	0.894819i	$0.647305\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	2.45714	0.237541	0.118770	−	0.992922i	$-0.462105\pi$
$107$	2.45714	0.237541	0.118770	+	0.992922i	$0.462105\pi$
$108$	−21.3723	−2.05655
$109$	−16.2865	−1.55996	−0.779981	−	0.625804i	$-0.784771\pi$
$109$	−16.2865	−1.55996	−0.779981	+	0.625804i	$0.784771\pi$
$110$	0	0
$111$	−25.6525	−2.43483
$112$	2.47214	0.233595
$113$	−2.80017	−0.263418	−0.131709	−	0.991288i	$-0.542046\pi$
$113$	−2.80017	−0.263418	−0.131709	+	0.991288i	$0.542046\pi$
$114$	0	0
$115$	0	0
$116$	−7.60845	−0.706427
$117$	−19.9192	−1.84153
$118$	0	0
$119$	1.76393	0.161699
$120$	0	0
$121$	23.2705	2.11550
$122$	0	0
$123$	35.6525	3.21468
$124$	−3.24920	−0.291787
$125$	0	0
$126$	0	0
$127$	−0.620541	−0.0550641	−0.0275321	−	0.999621i	$-0.508765\pi$
$127$	−0.620541	−0.0550641	−0.0275321	+	0.999621i	$0.508765\pi$
$128$	0	0
$129$	22.7194	2.00033
$130$	0	0
$131$	5.23607	0.457477	0.228739	−	0.973488i	$-0.426540\pi$
$131$	5.23607	0.457477	0.228739	+	0.973488i	$0.426540\pi$
$132$	36.0341	3.13637
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−14.5623	−1.24414	−0.622071	−	0.782961i	$-0.713708\pi$
$137$	−14.5623	−1.24414	−0.622071	+	0.782961i	$0.713708\pi$
$138$	0	0
$139$	−0.381966	−0.0323979	−0.0161990	−	0.999869i	$-0.505157\pi$
$139$	−0.381966	−0.0323979	−0.0161990	+	0.999869i	$0.505157\pi$
$140$	0	0
$141$	−14.4904	−1.22031
$142$	0	0
$143$	18.0171	1.50666
$144$	25.8885	2.15738
$145$	0	0
$146$	0	0
$147$	−20.3682	−1.67994
$148$	16.6700	1.37026
$149$	−3.76393	−0.308353	−0.154177	−	0.988043i	$-0.549272\pi$
$149$	−3.76393	−0.308353	−0.154177	+	0.988043i	$0.549272\pi$
$150$	0	0
$151$	3.35520	0.273042	0.136521	−	0.990637i	$-0.456408\pi$
$151$	3.35520	0.273042	0.136521	+	0.990637i	$0.456408\pi$
$152$	0	0
$153$	18.4721	1.49338
$154$	0	0
$155$	0	0
$156$	18.9443	1.51676
$157$	−15.0344	−1.19988	−0.599940	−	0.800045i	$-0.704809\pi$
$157$	−15.0344	−1.19988	−0.599940	+	0.800045i	$0.704809\pi$
$158$	0	0
$159$	−15.8541	−1.25731
$160$	0	0
$161$	−3.38197	−0.266536
$162$	0	0
$163$	12.0902	0.946975	0.473488	−	0.880800i	$-0.342995\pi$
$163$	12.0902	0.946975	0.473488	+	0.880800i	$0.342995\pi$
$164$	−23.1684	−1.80915
$165$	0	0
$166$	0	0
$167$	−2.45714	−0.190139	−0.0950697	−	0.995471i	$-0.530307\pi$
$167$	−2.45714	−0.190139	−0.0950697	+	0.995471i	$0.530307\pi$
$168$	0	0
$169$	−3.52786	−0.271374
$170$	0	0
$171$	0	0
$172$	−14.7639	−1.12574
$173$	−6.43288	−0.489083	−0.244541	−	0.969639i	$-0.578637\pi$
$173$	−6.43288	−0.489083	−0.244541	+	0.969639i	$0.578637\pi$
$174$	0	0
$175$	0	0
$176$	−23.4164	−1.76508
$177$	−13.0902	−0.983917
$178$	0	0
$179$	−5.42882	−0.405769	−0.202885	−	0.979203i	$-0.565032\pi$
$179$	−5.42882	−0.405769	−0.202885	+	0.979203i	$0.565032\pi$
$180$	0	0
$181$	−17.7396	−1.31857	−0.659286	−	0.751893i	$-0.729141\pi$
$181$	−17.7396	−1.31857	−0.659286	+	0.751893i	$0.729141\pi$
$182$	0	0
$183$	−22.2703	−1.64627
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−16.7082	−1.22182
$188$	9.41641	0.686762
$189$	6.60440	0.480399
$190$	0	0
$191$	6.09017	0.440669	0.220335	−	0.975424i	$-0.429285\pi$
$191$	6.09017	0.440669	0.220335	+	0.975424i	$0.429285\pi$
$192$	−24.6215	−1.77690
$193$	−23.4459	−1.68767	−0.843836	−	0.536601i	$-0.819708\pi$
$193$	−23.4459	−1.68767	−0.843836	+	0.536601i	$0.819708\pi$
$194$	0	0
$195$	0	0
$196$	13.2361	0.945433
$197$	8.65248	0.616463	0.308232	−	0.951311i	$-0.400263\pi$
$197$	8.65248	0.616463	0.308232	+	0.951311i	$0.400263\pi$
$198$	0	0
$199$	−7.56231	−0.536078	−0.268039	−	0.963408i	$-0.586376\pi$
$199$	−7.56231	−0.536078	−0.268039	+	0.963408i	$0.586376\pi$
$200$	0	0
$201$	22.5623	1.59142
$202$	0	0
$203$	2.35114	0.165018
$204$	−17.5680	−1.23001
$205$	0	0
$206$	0	0
$207$	−35.4164	−2.46161
$208$	−12.3107	−0.853596
$209$	0	0
$210$	0	0
$211$	−8.78402	−0.604717	−0.302359	−	0.953194i	$-0.597774\pi$
$211$	−8.78402	−0.604717	−0.302359	+	0.953194i	$0.597774\pi$
$212$	10.3026	0.707587
$213$	−0.527864	−0.0361686
$214$	0	0
$215$	0	0
$216$	0	0
$217$	1.00406	0.0681598
$218$	0	0
$219$	33.8545	2.28768
$220$	0	0
$221$	−8.78402	−0.590877
$222$	0	0
$223$	12.5882	0.842971	0.421486	−	0.906835i	$-0.361509\pi$
$223$	12.5882	0.842971	0.421486	+	0.906835i	$0.361509\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−9.61657	−0.638274	−0.319137	−	0.947709i	$-0.603393\pi$
$227$	−9.61657	−0.638274	−0.319137	+	0.947709i	$0.603393\pi$
$228$	0	0
$229$	1.85410	0.122523	0.0612613	−	0.998122i	$-0.480488\pi$
$229$	1.85410	0.122523	0.0612613	+	0.998122i	$0.480488\pi$
$230$	0	0
$231$	−11.1352	−0.732640
$232$	0	0
$233$	4.94427	0.323910	0.161955	−	0.986798i	$-0.448220\pi$
$233$	4.94427	0.323910	0.161955	+	0.986798i	$0.448220\pi$
$234$	0	0
$235$	0	0
$236$	8.50651	0.553727
$237$	−41.8328	−2.71733
$238$	0	0
$239$	−13.9443	−0.901980	−0.450990	−	0.892529i	$-0.648929\pi$
$239$	−13.9443	−0.901980	−0.450990	+	0.892529i	$0.648929\pi$
$240$	0	0
$241$	−3.18368	−0.205079	−0.102540	−	0.994729i	$-0.532697\pi$
$241$	−3.18368	−0.205079	−0.102540	+	0.994729i	$0.532697\pi$
$242$	0	0
$243$	9.40456	0.603303
$244$	14.4721	0.926484
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−32.9565	−2.08853
$250$	0	0
$251$	12.8541	0.811344	0.405672	−	0.914019i	$-0.367038\pi$
$251$	12.8541	0.811344	0.405672	+	0.914019i	$0.367038\pi$
$252$	−8.00000	−0.503953
$253$	32.0344	2.01399
$254$	0	0
$255$	0	0
$256$	16.0000	1.00000
$257$	5.36331	0.334554	0.167277	−	0.985910i	$-0.446503\pi$
$257$	5.36331	0.334554	0.167277	+	0.985910i	$0.446503\pi$
$258$	0	0
$259$	−5.15131	−0.320087
$260$	0	0
$261$	24.6215	1.52403
$262$	0	0
$263$	−22.0902	−1.36214	−0.681069	−	0.732219i	$-0.738485\pi$
$263$	−22.0902	−1.36214	−0.681069	+	0.732219i	$0.738485\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−25.3262	−1.54994
$268$	−14.6619	−0.895617
$269$	−22.3763	−1.36431	−0.682154	−	0.731208i	$-0.738957\pi$
$269$	−22.3763	−1.36431	−0.682154	+	0.731208i	$0.738957\pi$
$270$	0	0
$271$	8.41641	0.511260	0.255630	−	0.966775i	$-0.417717\pi$
$271$	8.41641	0.511260	0.255630	+	0.966775i	$0.417717\pi$
$272$	11.4164	0.692221
$273$	−5.85410	−0.354306
$274$	0	0
$275$	0	0
$276$	33.6830	2.02748
$277$	19.3607	1.16327	0.581635	−	0.813450i	$-0.302413\pi$
$277$	19.3607	1.16327	0.581635	+	0.813450i	$0.302413\pi$
$278$	0	0
$279$	10.5146	0.629494
$280$	0	0
$281$	15.6659	0.934551	0.467276	−	0.884112i	$-0.345236\pi$
$281$	15.6659	0.934551	0.467276	+	0.884112i	$0.345236\pi$
$282$	0	0
$283$	24.0000	1.42665	0.713326	−	0.700832i	$-0.247188\pi$
$283$	24.0000	1.42665	0.713326	+	0.700832i	$0.247188\pi$
$284$	0.343027	0.0203549
$285$	0	0
$286$	0	0
$287$	7.15942	0.422607
$288$	0	0
$289$	−8.85410	−0.520830
$290$	0	0
$291$	−25.6525	−1.50377
$292$	−22.0000	−1.28745
$293$	−22.5478	−1.31726	−0.658629	−	0.752467i	$-0.728864\pi$
$293$	−22.5478	−1.31726	−0.658629	+	0.752467i	$0.728864\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−62.5577	−3.62997
$298$	0	0
$299$	16.8415	0.973969
$300$	0	0
$301$	4.56231	0.262967
$302$	0	0
$303$	−5.87785	−0.337674
$304$	0	0
$305$	0	0
$306$	0	0
$307$	12.2047	0.696561	0.348280	−	0.937390i	$-0.386766\pi$
$307$	12.2047	0.696561	0.348280	+	0.937390i	$0.386766\pi$
$308$	7.23607	0.412313
$309$	−27.8885	−1.58652
$310$	0	0
$311$	−27.0344	−1.53298	−0.766491	−	0.642255i	$-0.777999\pi$
$311$	−27.0344	−1.53298	−0.766491	+	0.642255i	$0.777999\pi$
$312$	0	0
$313$	−13.6738	−0.772887	−0.386443	−	0.922313i	$-0.626297\pi$
$313$	−13.6738	−0.772887	−0.386443	+	0.922313i	$0.626297\pi$
$314$	0	0
$315$	0	0
$316$	27.1846	1.52925
$317$	34.4095	1.93263	0.966316	−	0.257357i	$-0.0828516\pi$
$317$	34.4095	1.93263	0.966316	+	0.257357i	$0.0828516\pi$
$318$	0	0
$319$	−22.2703	−1.24690
$320$	0	0
$321$	7.56231	0.422087
$322$	0	0
$323$	0	0
$324$	−26.9443	−1.49690
$325$	0	0
$326$	0	0
$327$	−50.1246	−2.77190
$328$	0	0
$329$	−2.90983	−0.160424
$330$	0	0
$331$	23.5519	1.29453	0.647265	−	0.762265i	$-0.275913\pi$
$331$	23.5519	1.29453	0.647265	+	0.762265i	$0.275913\pi$
$332$	21.4164	1.17538
$333$	−53.9452	−2.95618
$334$	0	0
$335$	0	0
$336$	7.60845	0.415075
$337$	−0.171513	−0.00934293	−0.00467147	−	0.999989i	$-0.501487\pi$
$337$	−0.171513	−0.00934293	−0.00467147	+	0.999989i	$0.501487\pi$
$338$	0	0
$339$	−8.61803	−0.468067
$340$	0	0
$341$	−9.51057	−0.515026
$342$	0	0
$343$	−8.41641	−0.454443
$344$	0	0
$345$	0	0
$346$	0	0
$347$	24.7082	1.32641	0.663203	−	0.748440i	$-0.269197\pi$
$347$	24.7082	1.32641	0.663203	+	0.748440i	$0.269197\pi$
$348$	−23.4164	−1.25525
$349$	10.1246	0.541958	0.270979	−	0.962585i	$-0.412653\pi$
$349$	10.1246	0.541958	0.270979	+	0.962585i	$0.412653\pi$
$350$	0	0
$351$	−32.8885	−1.75546
$352$	0	0
$353$	−15.3820	−0.818699	−0.409350	−	0.912378i	$-0.634244\pi$
$353$	−15.3820	−0.818699	−0.409350	+	0.912378i	$0.634244\pi$
$354$	0	0
$355$	0	0
$356$	16.4580	0.872272
$357$	5.42882	0.287324
$358$	0	0
$359$	−6.43769	−0.339768	−0.169884	−	0.985464i	$-0.554339\pi$
$359$	−6.43769	−0.339768	−0.169884	+	0.985464i	$0.554339\pi$
$360$	0	0
$361$	0	0
$362$	0	0
$363$	71.6193	3.75904
$364$	3.80423	0.199396
$365$	0	0
$366$	0	0
$367$	13.6525	0.712653	0.356327	−	0.934361i	$-0.384029\pi$
$367$	13.6525	0.712653	0.356327	+	0.934361i	$0.384029\pi$
$368$	−21.8885	−1.14102
$369$	74.9745	3.90301
$370$	0	0
$371$	−3.18368	−0.165289
$372$	−10.0000	−0.518476
$373$	34.9241	1.80830	0.904150	−	0.427214i	$-0.140505\pi$
$373$	34.9241	1.80830	0.904150	+	0.427214i	$0.140505\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−11.7082	−0.603003
$378$	0	0
$379$	30.7768	1.58090	0.790450	−	0.612527i	$-0.209847\pi$
$379$	30.7768	1.58090	0.790450	+	0.612527i	$0.209847\pi$
$380$	0	0
$381$	−1.90983	−0.0978436
$382$	0	0
$383$	21.2008	1.08331	0.541654	−	0.840601i	$-0.317798\pi$
$383$	21.2008	1.08331	0.541654	+	0.840601i	$0.317798\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	47.7771	2.42865
$388$	16.6700	0.846290
$389$	7.43769	0.377106	0.188553	−	0.982063i	$-0.439620\pi$
$389$	7.43769	0.377106	0.188553	+	0.982063i	$0.439620\pi$
$390$	0	0
$391$	−15.6180	−0.789838
$392$	0	0
$393$	16.1150	0.812892
$394$	0	0
$395$	0	0
$396$	75.7771	3.80794
$397$	16.4721	0.826713	0.413356	−	0.910569i	$-0.364356\pi$
$397$	16.4721	0.826713	0.413356	+	0.910569i	$0.364356\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	8.16348	0.407665	0.203832	−	0.979006i	$-0.434660\pi$
$401$	8.16348	0.407665	0.203832	+	0.979006i	$0.434660\pi$
$402$	0	0
$403$	−5.00000	−0.249068
$404$	3.81966	0.190035
$405$	0	0
$406$	0	0
$407$	48.7939	2.41862
$408$	0	0
$409$	−4.25325	−0.210310	−0.105155	−	0.994456i	$-0.533534\pi$
$409$	−4.25325	−0.210310	−0.105155	+	0.994456i	$0.533534\pi$
$410$	0	0
$411$	−44.8182	−2.21072
$412$	18.1231	0.892860
$413$	−2.62866	−0.129348
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−1.17557	−0.0575679
$418$	0	0
$419$	−23.2918	−1.13788	−0.568939	−	0.822379i	$-0.692646\pi$
$419$	−23.2918	−1.13788	−0.568939	+	0.822379i	$0.692646\pi$
$420$	0	0
$421$	16.8415	0.820805	0.410402	−	0.911905i	$-0.365388\pi$
$421$	16.8415	0.820805	0.410402	+	0.911905i	$0.365388\pi$
$422$	0	0
$423$	−30.4721	−1.48161
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−4.47214	−0.216422
$428$	−4.91428	−0.237541
$429$	55.4508	2.67719
$430$	0	0
$431$	−5.77185	−0.278020	−0.139010	−	0.990291i	$-0.544392\pi$
$431$	−5.77185	−0.278020	−0.139010	+	0.990291i	$0.544392\pi$
$432$	42.7445	2.05655
$433$	6.04937	0.290714	0.145357	−	0.989379i	$-0.453567\pi$
$433$	6.04937	0.290714	0.145357	+	0.989379i	$0.453567\pi$
$434$	0	0
$435$	0	0
$436$	32.5729	1.55996
$437$	0	0
$438$	0	0
$439$	−16.4985	−0.787429	−0.393715	−	0.919233i	$-0.628810\pi$
$439$	−16.4985	−0.787429	−0.393715	+	0.919233i	$0.628810\pi$
$440$	0	0
$441$	−42.8328	−2.03966
$442$	0	0
$443$	−17.9443	−0.852558	−0.426279	−	0.904592i	$-0.640176\pi$
$443$	−17.9443	−0.852558	−0.426279	+	0.904592i	$0.640176\pi$
$444$	51.3050	2.43483
$445$	0	0
$446$	0	0
$447$	−11.5842	−0.547913
$448$	−4.94427	−0.233595
$449$	−0.514540	−0.0242827	−0.0121413	−	0.999926i	$-0.503865\pi$
$449$	−0.514540	−0.0242827	−0.0121413	+	0.999926i	$0.503865\pi$
$450$	0	0
$451$	−67.8150	−3.19329
$452$	5.60034	0.263418
$453$	10.3262	0.485169
$454$	0	0
$455$	0	0
$456$	0	0
$457$	12.6525	0.591858	0.295929	−	0.955210i	$-0.404371\pi$
$457$	12.6525	0.591858	0.295929	+	0.955210i	$0.404371\pi$
$458$	0	0
$459$	30.4993	1.42359
$460$	0	0
$461$	−6.41641	−0.298842	−0.149421	−	0.988774i	$-0.547741\pi$
$461$	−6.41641	−0.298842	−0.149421	+	0.988774i	$0.547741\pi$
$462$	0	0
$463$	17.2918	0.803618	0.401809	−	0.915724i	$-0.368382\pi$
$463$	17.2918	0.803618	0.401809	+	0.915724i	$0.368382\pi$
$464$	15.2169	0.706427
$465$	0	0
$466$	0	0
$467$	−10.9443	−0.506441	−0.253220	−	0.967409i	$-0.581490\pi$
$467$	−10.9443	−0.506441	−0.253220	+	0.967409i	$0.581490\pi$
$468$	39.8384	1.84153
$469$	4.53077	0.209211
$470$	0	0
$471$	−46.2713	−2.13207
$472$	0	0
$473$	−43.2148	−1.98702
$474$	0	0
$475$	0	0
$476$	−3.52786	−0.161699
$477$	−33.3400	−1.52653
$478$	0	0
$479$	25.1459	1.14895	0.574473	−	0.818524i	$-0.305207\pi$
$479$	25.1459	1.14895	0.574473	+	0.818524i	$0.305207\pi$
$480$	0	0
$481$	25.6525	1.16965
$482$	0	0
$483$	−10.4086	−0.473609
$484$	−46.5410	−2.11550
$485$	0	0
$486$	0	0
$487$	−19.9192	−0.902624	−0.451312	−	0.892366i	$-0.649044\pi$
$487$	−19.9192	−0.902624	−0.451312	+	0.892366i	$0.649044\pi$
$488$	0	0
$489$	37.2097	1.68268
$490$	0	0
$491$	−30.6180	−1.38177	−0.690886	−	0.722963i	$-0.742779\pi$
$491$	−30.6180	−1.38177	−0.690886	+	0.722963i	$0.742779\pi$
$492$	−71.3050	−3.21468
$493$	10.8576	0.489004
$494$	0	0
$495$	0	0
$496$	6.49839	0.291787
$497$	−0.106001	−0.00475480
$498$	0	0
$499$	16.1246	0.721837	0.360918	−	0.932597i	$-0.382463\pi$
$499$	16.1246	0.721837	0.360918	+	0.932597i	$0.382463\pi$
$500$	0	0
$501$	−7.56231	−0.337859
$502$	0	0
$503$	−25.1459	−1.12120	−0.560600	−	0.828087i	$-0.689429\pi$
$503$	−25.1459	−1.12120	−0.560600	+	0.828087i	$0.689429\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−10.8576	−0.482205
$508$	1.24108	0.0550641
$509$	−22.2048	−0.984211	−0.492106	−	0.870536i	$-0.663773\pi$
$509$	−22.2048	−0.984211	−0.492106	+	0.870536i	$0.663773\pi$
$510$	0	0
$511$	6.79837	0.300742
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	−45.4387	−2.00033
$517$	27.5623	1.21219
$518$	0	0
$519$	−19.7984	−0.869052
$520$	0	0
$521$	41.7405	1.82868	0.914342	−	0.404943i	$-0.132709\pi$
$521$	41.7405	1.82868	0.914342	+	0.404943i	$0.132709\pi$
$522$	0	0
$523$	−23.1684	−1.01308	−0.506541	−	0.862216i	$-0.669076\pi$
$523$	−23.1684	−1.01308	−0.506541	+	0.862216i	$0.669076\pi$
$524$	−10.4721	−0.457477
$525$	0	0
$526$	0	0
$527$	4.63677	0.201981
$528$	−72.0683	−3.13637
$529$	6.94427	0.301925
$530$	0	0
$531$	−27.5276	−1.19460
$532$	0	0
$533$	−35.6525	−1.54428
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−16.7082	−0.721012
$538$	0	0
$539$	38.7426	1.66876
$540$	0	0
$541$	−28.9443	−1.24441	−0.622206	−	0.782854i	$-0.713763\pi$
$541$	−28.9443	−1.24441	−0.622206	+	0.782854i	$0.713763\pi$
$542$	0	0
$543$	−54.5967	−2.34297
$544$	0	0
$545$	0	0
$546$	0	0
$547$	26.9726	1.15327	0.576633	−	0.817003i	$-0.304366\pi$
$547$	26.9726	1.15327	0.576633	+	0.817003i	$0.304366\pi$
$548$	29.1246	1.24414
$549$	−46.8328	−1.99878
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−8.40051	−0.357226
$554$	0	0
$555$	0	0
$556$	0.763932	0.0323979
$557$	34.3607	1.45591	0.727954	−	0.685626i	$-0.240471\pi$
$557$	34.3607	1.45591	0.727954	+	0.685626i	$0.240471\pi$
$558$	0	0
$559$	−22.7194	−0.960926
$560$	0	0
$561$	−51.4226	−2.17106
$562$	0	0
$563$	11.5842	0.488215	0.244108	−	0.969748i	$-0.421505\pi$
$563$	11.5842	0.488215	0.244108	+	0.969748i	$0.421505\pi$
$564$	28.9807	1.22031
$565$	0	0
$566$	0	0
$567$	8.32624	0.349669
$568$	0	0
$569$	−36.5892	−1.53390	−0.766949	−	0.641708i	$-0.778226\pi$
$569$	−36.5892	−1.53390	−0.766949	+	0.641708i	$0.778226\pi$
$570$	0	0
$571$	−38.5410	−1.61289	−0.806446	−	0.591308i	$-0.798612\pi$
$571$	−38.5410	−1.61289	−0.806446	+	0.591308i	$0.798612\pi$
$572$	−36.0341	−1.50666
$573$	18.7436	0.783026
$574$	0	0
$575$	0	0
$576$	−51.7771	−2.15738
$577$	12.1246	0.504754	0.252377	−	0.967629i	$-0.418788\pi$
$577$	12.1246	0.504754	0.252377	+	0.967629i	$0.418788\pi$
$578$	0	0
$579$	−72.1591	−2.99883
$580$	0	0
$581$	−6.61803	−0.274562
$582$	0	0
$583$	30.1563	1.24895
$584$	0	0
$585$	0	0
$586$	0	0
$587$	19.8885	0.820888	0.410444	−	0.911886i	$-0.365374\pi$
$587$	19.8885	0.820888	0.410444	+	0.911886i	$0.365374\pi$
$588$	40.7364	1.67994
$589$	0	0
$590$	0	0
$591$	26.6296	1.09539
$592$	−33.3400	−1.37026
$593$	24.2148	0.994382	0.497191	−	0.867641i	$-0.334365\pi$
$593$	24.2148	0.994382	0.497191	+	0.867641i	$0.334365\pi$
$594$	0	0
$595$	0	0
$596$	7.52786	0.308353
$597$	−23.2744	−0.952557
$598$	0	0
$599$	−27.2501	−1.11341	−0.556705	−	0.830710i	$-0.687935\pi$
$599$	−27.2501	−1.11341	−0.556705	+	0.830710i	$0.687935\pi$
$600$	0	0
$601$	36.0341	1.46986	0.734932	−	0.678141i	$-0.237214\pi$
$601$	36.0341	1.46986	0.734932	+	0.678141i	$0.237214\pi$
$602$	0	0
$603$	47.4468	1.93218
$604$	−6.71040	−0.273042
$605$	0	0
$606$	0	0
$607$	42.5325	1.72634	0.863171	−	0.504911i	$-0.168475\pi$
$607$	42.5325	1.72634	0.863171	+	0.504911i	$0.168475\pi$
$608$	0	0
$609$	7.23607	0.293220
$610$	0	0
$611$	14.4904	0.586217
$612$	−36.9443	−1.49338
$613$	23.2705	0.939887	0.469944	−	0.882696i	$-0.344274\pi$
$613$	23.2705	0.939887	0.469944	+	0.882696i	$0.344274\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−0.888544	−0.0357714	−0.0178857	−	0.999840i	$-0.505694\pi$
$617$	−0.888544	−0.0357714	−0.0178857	+	0.999840i	$0.505694\pi$
$618$	0	0
$619$	−20.9098	−0.840437	−0.420219	−	0.907423i	$-0.638047\pi$
$619$	−20.9098	−0.840437	−0.420219	+	0.907423i	$0.638047\pi$
$620$	0	0
$621$	−58.4760	−2.34656
$622$	0	0
$623$	−5.08580	−0.203758
$624$	−37.8885	−1.51676
$625$	0	0
$626$	0	0
$627$	0	0
$628$	30.0689	1.19988
$629$	−23.7889	−0.948527
$630$	0	0
$631$	−24.4721	−0.974220	−0.487110	−	0.873341i	$-0.661949\pi$
$631$	−24.4721	−0.974220	−0.487110	+	0.873341i	$0.661949\pi$
$632$	0	0
$633$	−27.0344	−1.07452
$634$	0	0
$635$	0	0
$636$	31.7082	1.25731
$637$	20.3682	0.807018
$638$	0	0
$639$	−1.11006	−0.0439132
$640$	0	0
$641$	−19.3642	−0.764838	−0.382419	−	0.923989i	$-0.624909\pi$
$641$	−19.3642	−0.764838	−0.382419	+	0.923989i	$0.624909\pi$
$642$	0	0
$643$	−14.9443	−0.589345	−0.294672	−	0.955598i	$-0.595210\pi$
$643$	−14.9443	−0.589345	−0.294672	+	0.955598i	$0.595210\pi$
$644$	6.76393	0.266536
$645$	0	0
$646$	0	0
$647$	−45.3607	−1.78331	−0.891656	−	0.452713i	$-0.850456\pi$
$647$	−45.3607	−1.78331	−0.891656	+	0.452713i	$0.850456\pi$
$648$	0	0
$649$	24.8990	0.977371
$650$	0	0
$651$	3.09017	0.121113
$652$	−24.1803	−0.946975
$653$	9.32624	0.364964	0.182482	−	0.983209i	$-0.441587\pi$
$653$	9.32624	0.364964	0.182482	+	0.983209i	$0.441587\pi$
$654$	0	0
$655$	0	0
$656$	46.3368	1.80915
$657$	71.1935	2.77752
$658$	0	0
$659$	29.6013	1.15310	0.576551	−	0.817061i	$-0.304398\pi$
$659$	29.6013	1.15310	0.576551	+	0.817061i	$0.304398\pi$
$660$	0	0
$661$	20.0907	0.781438	0.390719	−	0.920510i	$-0.372226\pi$
$661$	20.0907	0.781438	0.390719	+	0.920510i	$0.372226\pi$
$662$	0	0
$663$	−27.0344	−1.04993
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−20.8172	−0.806047
$668$	4.91428	0.190139
$669$	38.7426	1.49788
$670$	0	0
$671$	42.3607	1.63532
$672$	0	0
$673$	0.661030	0.0254808	0.0127404	−	0.999919i	$-0.495944\pi$
$673$	0.661030	0.0254808	0.0127404	+	0.999919i	$0.495944\pi$
$674$	0	0
$675$	0	0
$676$	7.05573	0.271374
$677$	−4.04125	−0.155318	−0.0776590	−	0.996980i	$-0.524745\pi$
$677$	−4.04125	−0.155318	−0.0776590	+	0.996980i	$0.524745\pi$
$678$	0	0
$679$	−5.15131	−0.197689
$680$	0	0
$681$	−29.5967	−1.13415
$682$	0	0
$683$	−26.8011	−1.02552	−0.512758	−	0.858533i	$-0.671376\pi$
$683$	−26.8011	−1.02552	−0.512758	+	0.858533i	$0.671376\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	5.70634	0.217710
$688$	29.5279	1.12574
$689$	15.8541	0.603993
$690$	0	0
$691$	−28.9443	−1.10109	−0.550546	−	0.834805i	$-0.685580\pi$
$691$	−28.9443	−1.10109	−0.550546	+	0.834805i	$0.685580\pi$
$692$	12.8658	0.489083
$693$	−23.4164	−0.889516
$694$	0	0
$695$	0	0
$696$	0	0
$697$	33.0625	1.25233
$698$	0	0
$699$	15.2169	0.575556
$700$	0	0
$701$	25.0000	0.944237	0.472118	−	0.881535i	$-0.343489\pi$
$701$	25.0000	0.944237	0.472118	+	0.881535i	$0.343489\pi$
$702$	0	0
$703$	0	0
$704$	46.8328	1.76508
$705$	0	0
$706$	0	0
$707$	−1.18034	−0.0443913
$708$	26.1803	0.983917
$709$	−25.5279	−0.958719	−0.479360	−	0.877619i	$-0.659131\pi$
$709$	−25.5279	−0.958719	−0.479360	+	0.877619i	$0.659131\pi$
$710$	0	0
$711$	−87.9713	−3.29918
$712$	0	0
$713$	−8.89002	−0.332934
$714$	0	0
$715$	0	0
$716$	10.8576	0.405769
$717$	−42.9161	−1.60273
$718$	0	0
$719$	8.23607	0.307154	0.153577	−	0.988137i	$-0.450921\pi$
$719$	8.23607	0.307154	0.153577	+	0.988137i	$0.450921\pi$
$720$	0	0
$721$	−5.60034	−0.208567
$722$	0	0
$723$	−9.79837	−0.364405
$724$	35.4791	1.31857
$725$	0	0
$726$	0	0
$727$	−21.0689	−0.781402	−0.390701	−	0.920518i	$-0.627767\pi$
$727$	−21.0689	−0.781402	−0.390701	+	0.920518i	$0.627767\pi$
$728$	0	0
$729$	−11.4721	−0.424894
$730$	0	0
$731$	21.0689	0.779261
$732$	44.5407	1.64627
$733$	25.7984	0.952885	0.476442	−	0.879206i	$-0.341926\pi$
$733$	25.7984	0.952885	0.476442	+	0.879206i	$0.341926\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−42.9161	−1.58083
$738$	0	0
$739$	−38.4164	−1.41317	−0.706585	−	0.707628i	$-0.749765\pi$
$739$	−38.4164	−1.41317	−0.706585	+	0.707628i	$0.749765\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	38.4508	1.41062	0.705312	−	0.708897i	$-0.250807\pi$
$743$	38.4508	1.41062	0.705312	+	0.708897i	$0.250807\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−69.3050	−2.53574
$748$	33.4164	1.22182
$749$	1.51860	0.0554883
$750$	0	0
$751$	6.53888	0.238607	0.119304	−	0.992858i	$-0.461934\pi$
$751$	6.53888	0.238607	0.119304	+	0.992858i	$0.461934\pi$
$752$	−18.8328	−0.686762
$753$	39.5609	1.44168
$754$	0	0
$755$	0	0
$756$	−13.2088	−0.480399
$757$	−52.7771	−1.91822	−0.959108	−	0.283041i	$-0.908657\pi$
$757$	−52.7771	−1.91822	−0.959108	+	0.283041i	$0.908657\pi$
$758$	0	0
$759$	98.5919	3.57866
$760$	0	0
$761$	−39.0689	−1.41625	−0.708123	−	0.706089i	$-0.750458\pi$
$761$	−39.0689	−1.41625	−0.708123	+	0.706089i	$0.750458\pi$
$762$	0	0
$763$	−10.0656	−0.364399
$764$	−12.1803	−0.440669
$765$	0	0
$766$	0	0
$767$	13.0902	0.472659
$768$	49.2429	1.77690
$769$	18.3607	0.662103	0.331052	−	0.943613i	$-0.392597\pi$
$769$	18.3607	0.662103	0.331052	+	0.943613i	$0.392597\pi$
$770$	0	0
$771$	16.5066	0.594470
$772$	46.8918	1.68767
$773$	−21.3723	−0.768707	−0.384354	−	0.923186i	$-0.625576\pi$
$773$	−21.3723	−0.768707	−0.384354	+	0.923186i	$0.625576\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−15.8541	−0.568763
$778$	0	0
$779$	0	0
$780$	0	0
$781$	1.00406	0.0359280
$782$	0	0
$783$	40.6525	1.45280
$784$	−26.4721	−0.945433
$785$	0	0
$786$	0	0
$787$	−22.0583	−0.786294	−0.393147	−	0.919476i	$-0.628614\pi$
$787$	−22.0583	−0.786294	−0.393147	+	0.919476i	$0.628614\pi$
$788$	−17.3050	−0.616463
$789$	−67.9866	−2.42039
$790$	0	0
$791$	−1.73060	−0.0615330
$792$	0	0
$793$	22.2703	0.790843
$794$	0	0
$795$	0	0
$796$	15.1246	0.536078
$797$	12.4822	0.442144	0.221072	−	0.975258i	$-0.429044\pi$
$797$	12.4822	0.442144	0.221072	+	0.975258i	$0.429044\pi$
$798$	0	0
$799$	−13.4377	−0.475391
$800$	0	0
$801$	−53.2592	−1.88182
$802$	0	0
$803$	−64.3951	−2.27245
$804$	−45.1246	−1.59142
$805$	0	0
$806$	0	0
$807$	−68.8673	−2.42424
$808$	0	0
$809$	42.2361	1.48494	0.742471	−	0.669879i	$-0.233654\pi$
$809$	42.2361	1.48494	0.742471	+	0.669879i	$0.233654\pi$
$810$	0	0
$811$	31.9524	1.12200	0.561000	−	0.827816i	$-0.310417\pi$
$811$	31.9524	1.12200	0.561000	+	0.827816i	$0.310417\pi$
$812$	−4.70228	−0.165018
$813$	25.9030	0.908459
$814$	0	0
$815$	0	0
$816$	35.1361	1.23001
$817$	0	0
$818$	0	0
$819$	−12.3107	−0.430172
$820$	0	0
$821$	17.5967	0.614131	0.307065	−	0.951688i	$-0.400653\pi$
$821$	17.5967	0.614131	0.307065	+	0.951688i	$0.400653\pi$
$822$	0	0
$823$	−18.8885	−0.658413	−0.329207	−	0.944258i	$-0.606781\pi$
$823$	−18.8885	−0.658413	−0.329207	+	0.944258i	$0.606781\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−25.0705	−0.871787	−0.435893	−	0.899998i	$-0.643567\pi$
$827$	−25.0705	−0.871787	−0.435893	+	0.899998i	$0.643567\pi$
$828$	70.8328	2.46161
$829$	32.0584	1.11343	0.556717	−	0.830702i	$-0.312061\pi$
$829$	32.0584	1.11343	0.556717	+	0.830702i	$0.312061\pi$
$830$	0	0
$831$	59.5860	2.06702
$832$	24.6215	0.853596
$833$	−18.8885	−0.654449
$834$	0	0
$835$	0	0
$836$	0	0
$837$	17.3607	0.600073
$838$	0	0
$839$	24.7930	0.855949	0.427974	−	0.903791i	$-0.359227\pi$
$839$	24.7930	0.855949	0.427974	+	0.903791i	$0.359227\pi$
$840$	0	0
$841$	−14.5279	−0.500961
$842$	0	0
$843$	48.2148	1.66061
$844$	17.5680	0.604717
$845$	0	0
$846$	0	0
$847$	14.3820	0.494170
$848$	−20.6052	−0.707587
$849$	73.8644	2.53502
$850$	0	0
$851$	45.6102	1.56350
$852$	1.05573	0.0361686
$853$	30.2705	1.03644	0.518221	−	0.855247i	$-0.326594\pi$
$853$	30.2705	1.03644	0.518221	+	0.855247i	$0.326594\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	47.1693	1.61127	0.805636	−	0.592410i	$-0.201824\pi$
$857$	47.1693	1.61127	0.805636	+	0.592410i	$0.201824\pi$
$858$	0	0
$859$	6.83282	0.233133	0.116566	−	0.993183i	$-0.462811\pi$
$859$	6.83282	0.233133	0.116566	+	0.993183i	$0.462811\pi$
$860$	0	0
$861$	22.0344	0.750932
$862$	0	0
$863$	9.02105	0.307080	0.153540	−	0.988142i	$-0.450933\pi$
$863$	9.02105	0.307080	0.153540	+	0.988142i	$0.450933\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−27.2501	−0.925463
$868$	−2.00811	−0.0681598
$869$	79.5707	2.69925
$870$	0	0
$871$	−22.5623	−0.764495
$872$	0	0
$873$	−53.9452	−1.82577
$874$	0	0
$875$	0	0
$876$	−67.7090	−2.28768
$877$	26.4581	0.893426	0.446713	−	0.894677i	$-0.352595\pi$
$877$	26.4581	0.893426	0.446713	+	0.894677i	$0.352595\pi$
$878$	0	0
$879$	−69.3951	−2.34064
$880$	0	0
$881$	29.5967	0.997140	0.498570	−	0.866850i	$-0.333859\pi$
$881$	29.5967	0.997140	0.498570	+	0.866850i	$0.333859\pi$
$882$	0	0
$883$	32.0902	1.07992	0.539960	−	0.841691i	$-0.318439\pi$
$883$	32.0902	1.07992	0.539960	+	0.841691i	$0.318439\pi$
$884$	17.5680	0.590877
$885$	0	0
$886$	0	0
$887$	−58.7940	−1.97411	−0.987055	−	0.160385i	$-0.948726\pi$
$887$	−58.7940	−1.97411	−0.987055	+	0.160385i	$0.948726\pi$
$888$	0	0
$889$	−0.383516	−0.0128627
$890$	0	0
$891$	−78.8673	−2.64215
$892$	−25.1765	−0.842971
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	51.8328	1.73065
$898$	0	0
$899$	6.18034	0.206126
$900$	0	0
$901$	−14.7024	−0.489807
$902$	0	0
$903$	14.0413	0.467266
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−36.5237	−1.21275	−0.606374	−	0.795179i	$-0.707377\pi$
$907$	−36.5237	−1.21275	−0.606374	+	0.795179i	$0.707377\pi$
$908$	19.2331	0.638274
$909$	−12.3607	−0.409978
$910$	0	0
$911$	−41.0139	−1.35885	−0.679426	−	0.733744i	$-0.737771\pi$
$911$	−41.0139	−1.35885	−0.679426	+	0.733744i	$0.737771\pi$
$912$	0	0
$913$	62.6869	2.07463
$914$	0	0
$915$	0	0
$916$	−3.70820	−0.122523
$917$	3.23607	0.106864
$918$	0	0
$919$	9.59675	0.316567	0.158284	−	0.987394i	$-0.449404\pi$
$919$	9.59675	0.316567	0.158284	+	0.987394i	$0.449404\pi$
$920$	0	0
$921$	37.5623	1.23772
$922$	0	0
$923$	0.527864	0.0173749
$924$	22.2703	0.732640
$925$	0	0
$926$	0	0
$927$	−58.6475	−1.92624
$928$	0	0
$929$	−23.5410	−0.772356	−0.386178	−	0.922424i	$-0.626205\pi$
$929$	−23.5410	−0.772356	−0.386178	+	0.922424i	$0.626205\pi$
$930$	0	0
$931$	0	0
$932$	−9.88854	−0.323910
$933$	−83.2035	−2.72396
$934$	0	0
$935$	0	0
$936$	0	0
$937$	19.4377	0.635002	0.317501	−	0.948258i	$-0.397156\pi$
$937$	19.4377	0.635002	0.317501	+	0.948258i	$0.397156\pi$
$938$	0	0
$939$	−42.0835	−1.37334
$940$	0	0
$941$	−4.49028	−0.146379	−0.0731895	−	0.997318i	$-0.523318\pi$
$941$	−4.49028	−0.146379	−0.0731895	+	0.997318i	$0.523318\pi$
$942$	0	0
$943$	−63.3903	−2.06427
$944$	−17.0130	−0.553727
$945$	0	0
$946$	0	0
$947$	38.1033	1.23819	0.619096	−	0.785315i	$-0.287499\pi$
$947$	38.1033	1.23819	0.619096	+	0.785315i	$0.287499\pi$
$948$	83.6656	2.71733
$949$	−33.8545	−1.09896
$950$	0	0
$951$	105.902	3.43410
$952$	0	0
$953$	23.3804	0.757365	0.378682	−	0.925527i	$-0.376377\pi$
$953$	23.3804	0.757365	0.378682	+	0.925527i	$0.376377\pi$
$954$	0	0
$955$	0	0
$956$	27.8885	0.901980
$957$	−68.5410	−2.21562
$958$	0	0
$959$	−9.00000	−0.290625
$960$	0	0
$961$	−28.3607	−0.914861
$962$	0	0
$963$	15.9030	0.512466
$964$	6.36737	0.205079
$965$	0	0
$966$	0	0
$967$	42.0689	1.35284	0.676422	−	0.736514i	$-0.263530\pi$
$967$	42.0689	1.35284	0.676422	+	0.736514i	$0.263530\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−21.8618	−0.701578	−0.350789	−	0.936454i	$-0.614087\pi$
$971$	−21.8618	−0.701578	−0.350789	+	0.936454i	$0.614087\pi$
$972$	−18.8091	−0.603303
$973$	−0.236068	−0.00756799
$974$	0	0
$975$	0	0
$976$	−28.9443	−0.926484
$977$	32.8505	1.05098	0.525490	−	0.850800i	$-0.323882\pi$
$977$	32.8505	1.05098	0.525490	+	0.850800i	$0.323882\pi$
$978$	0	0
$979$	48.1734	1.53963
$980$	0	0
$981$	−105.408	−3.36543
$982$	0	0
$983$	5.81234	0.185385	0.0926924	−	0.995695i	$-0.470453\pi$
$983$	5.81234	0.185385	0.0926924	+	0.995695i	$0.470453\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−8.95554	−0.285058
$988$	0	0
$989$	−40.3951	−1.28449
$990$	0	0
$991$	30.6708	0.974291	0.487146	−	0.873321i	$-0.338038\pi$
$991$	30.6708	0.974291	0.487146	+	0.873321i	$0.338038\pi$
$992$	0	0
$993$	72.4853	2.30025
$994$	0	0
$995$	0	0
$996$	65.9129	2.08853
$997$	−9.23607	−0.292509	−0.146255	−	0.989247i	$-0.546722\pi$
$997$	−9.23607	−0.292509	−0.146255	+	0.989247i	$0.546722\pi$
$998$	0	0
$999$	−89.0689	−2.81801

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9025.2.a.bi.1.4		4
5.4	even	2		1805.2.a.k.1.1	✓	4
19.18	odd	2	inner	9025.2.a.bi.1.1		4
95.94	odd	2		1805.2.a.k.1.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1805.2.a.k.1.1	✓	4	5.4	even	2
1805.2.a.k.1.4	yes	4	95.94	odd	2
9025.2.a.bi.1.1		4	19.18	odd	2	inner
9025.2.a.bi.1.4		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 9025 = 5^{2} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9025.a (trivial)

\(f(q)\)	\(=\)	\(q+3.07768 q^{3} -2.00000 q^{4} +0.618034 q^{7} +6.47214 q^{9} +O(q^{10})\)	\(q+3.07768 q^{3} -2.00000 q^{4} +0.618034 q^{7} +6.47214 q^{9} -5.85410 q^{11} -6.15537 q^{12} -3.07768 q^{13} +4.00000 q^{16} +2.85410 q^{17} +1.90211 q^{21} -5.47214 q^{23} +10.6861 q^{27} -1.23607 q^{28} +3.80423 q^{29} +1.62460 q^{31} -18.0171 q^{33} -12.9443 q^{36} -8.33499 q^{37} -9.47214 q^{39} +11.5842 q^{41} +7.38197 q^{43} +11.7082 q^{44} -4.70820 q^{47} +12.3107 q^{48} -6.61803 q^{49} +8.78402 q^{51} +6.15537 q^{52} -5.15131 q^{53} -4.25325 q^{59} -7.23607 q^{61} +4.00000 q^{63} -8.00000 q^{64} +7.33094 q^{67} -5.70820 q^{68} -16.8415 q^{69} -0.171513 q^{71} +11.0000 q^{73} -3.61803 q^{77} -13.5923 q^{79} +13.4721 q^{81} -10.7082 q^{83} -3.80423 q^{84} +11.7082 q^{87} -8.22899 q^{89} -1.90211 q^{91} +10.9443 q^{92} +5.00000 q^{93} -8.33499 q^{97} -37.8885 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{4} - 2 q^{7} + 8 q^{9}+O(q^{10}) \) 4 * q - 8 * q^4 - 2 * q^7 + 8 * q^9	\( 4 q - 8 q^{4} - 2 q^{7} + 8 q^{9} - 10 q^{11} + 16 q^{16} - 2 q^{17} - 4 q^{23} + 4 q^{28} - 16 q^{36} - 20 q^{39} + 34 q^{43} + 20 q^{44} + 8 q^{47} - 22 q^{49} - 20 q^{61} + 16 q^{63} - 32 q^{64} + 4 q^{68} + 44 q^{73} - 10 q^{77} + 36 q^{81} - 16 q^{83} + 20 q^{87} + 8 q^{92} + 20 q^{93} - 80 q^{99}+O(q^{100}) \) 4 * q - 8 * q^4 - 2 * q^7 + 8 * q^9 - 10 * q^11 + 16 * q^16 - 2 * q^17 - 4 * q^23 + 4 * q^28 - 16 * q^36 - 20 * q^39 + 34 * q^43 + 20 * q^44 + 8 * q^47 - 22 * q^49 - 20 * q^61 + 16 * q^63 - 32 * q^64 + 4 * q^68 + 44 * q^73 - 10 * q^77 + 36 * q^81 - 16 * q^83 + 20 * q^87 + 8 * q^92 + 20 * q^93 - 80 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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